Even though over
1,000,000 digits of this number have been calculated, it is
not yet known if it is a rational number (the ratio of two
integers a/b). But if it is rational, the
denominator (b) must have more than 244,663 digits!

The constant gamma occurs in many places in number
theory; we will give three examples. First, de la Vallée
Poussin proved the following in 1898:

Take any positive integer n and divide it by
each positive integer m less than n.
Calculate the average (mean) fraction by which the
quotient n/m falls short of the
next integer.
The larger n gets, the closer the average
gets to gamma (not 1/2!)